Average Error: 45.8 → 0.0
Time: 45.1s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 226.96535516399638:\\ \;\;\;\;\frac{\frac{i}{4} \cdot i}{\frac{{\left(i \cdot \left(4 \cdot i\right)\right)}^{3} - {1.0}^{3}}{\left(i \cdot \left(4 \cdot i\right)\right) \cdot \left(i \cdot \left(4 \cdot i\right)\right) + \left(1.0 \cdot 1.0 + \left(i \cdot \left(4 \cdot i\right)\right) \cdot 1.0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right) + \frac{\frac{0.015625}{i}}{i}\\ \end{array}\]

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 226.96535516399638

    1. Initial program 44.9

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Initial simplification0.0

      \[\leadsto \frac{i \cdot \frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}\]
    3. Using strategy rm

    4. Applied flip3--0.0

      \[\leadsto \frac{i \cdot \frac{i}{4}}{\color{blue}{\frac{{\left(i \cdot \left(4 \cdot i\right)\right)}^{3} - {1.0}^{3}}{\left(i \cdot \left(4 \cdot i\right)\right) \cdot \left(i \cdot \left(4 \cdot i\right)\right) + \left(1.0 \cdot 1.0 + \left(i \cdot \left(4 \cdot i\right)\right) \cdot 1.0\right)}}}\]

    if 226.96535516399638 < i

    1. Initial program 46.7

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Initial simplification31.5

      \[\leadsto \frac{i \cdot \frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\frac{\frac{0.015625}{i}}{i} + \left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 226.96535516399638:\\ \;\;\;\;\frac{\frac{i}{4} \cdot i}{\frac{{\left(i \cdot \left(4 \cdot i\right)\right)}^{3} - {1.0}^{3}}{\left(i \cdot \left(4 \cdot i\right)\right) \cdot \left(i \cdot \left(4 \cdot i\right)\right) + \left(1.0 \cdot 1.0 + \left(i \cdot \left(4 \cdot i\right)\right) \cdot 1.0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right) + \frac{\frac{0.015625}{i}}{i}\\ \end{array}\]

Runtime

Time bar (total: 45.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes26.20.00.026.2100%
herbie shell --seed 2018354 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))